Evolution PDEs and augmented eigenfunctions. I finite interval

Smith, David A and Fokas, Athanassios S (2013) Evolution PDEs and augmented eigenfunctions. I finite interval. (Submitted)

[img] Text
acmac-0167.pdf

Download (395Kb)

Abstract

The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type~II, there does \emph{not} exist a classical transform pair and the solution \emph{cannot} be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors.

Item Type: Article
Uncontrolled Keywords: Linear differential operator, Initial-boundary value problem, Completeness of generalised eigenfunctions
Subjects: Q Science > QA Mathematics
Depositing User: Dr David A Smith
Date Deposited: 11 Mar 2013 11:21
Last Modified: 19 Nov 2017 23:41
URI: http://preprints.acmac.uoc.gr/id/eprint/167

Actions (login required)

View Item View Item